The question is: What is arc length anyway?
One possible definition is: For $\epsilon>0$, consider all point sequences $a=x_0,x_1,\ldots, x_n=b$ such that all $x_i$ are on the circle (or in any other set $S$ with respect to which we want to measure an arc length) and the distance $d(x_i,x_{i+1})$ between $x_i$ and $x_{i+1}$ is $<\epsilon$. Let $d_\epsilon(a,b)$ be the infimum of $d(x_0,x_1)+d(x_1,x_2)+\ldots +d(x_{n-1},x_n)$ over all such sequences. Then the arc length from $a$ to $b$ can be defined as $\lim_{\epsilon\to 0} d_\epsilon(a,b)$.
By the triangle inequality, $$d(x_0,x_1)+d(x_1,x_2)+\ldots +d(x_{n-1},x_n)\ge d(x_0,x_n)=d(a,b)$$ for all considered point sequences. Hence $d_\epsilon(a,b)\ge d(a,b)$ for all $\epsilon$, and hence the same inequality holds for the limit.
To show that the arc length is in fact strictly greater, pick any point $c$ on the arc between $a$ and $b$, and note that all point sequences with step width $<\epsilon$ have an intermediate point $x_i$ close to $c$ (that is: with $d(x_i,c)<\epsilon$). Hence in the limit we obtain that the arc length from $a$ to $b$ is $\ge d(a,c)+d(c,b)>d(a,b)$, where the last strict inequality follows from the fact that $c$ is not on the straight line segment $ab$.
